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Statistical inference for Gibbs point processes based on field observations

  • C. ComasEmail author
  • J. Mateu
Original Paper

Abstract

Forest inventories are mostly based on field observations, and complete records of spatial tree coordinates are seldom taken. The lack of individual coordinates prevents the use of well stablised statistical inference tools based on the likelihood function. However, the Takacs–Fiksel approach, based on equating two expectations derived from different measures, can be used routinely without any measurement of tree coordinates, just by considering nearest neighbour measurements and the counting of trees at some random positions. Despite this, little attention has been paid to the Takacs–Fiksel method in terms of the type of test function and the type of field observation data considered. Motivated by problems based on field observations, we present a simulation study to analyse and illustrate the quality of the parameter estimates for this estimation approach under distinct simulated scenarios, where several test functions and distinct forest sampling designs are taken into account. Indeed, the type of the chosen test function affects the resulting estimates in terms of the forest field observation considered.

Keywords

Forest field observations Forest sampling Monte Carlo simulation Nearest neighbour measurements Pairwise interaction point processes Spatial point patterns Takacs–Fiksel method 

Notes

Acknowledgements

We are grateful to the Editor, AE and two anonymous referees whose comments and suggestions have clearly improved an earlier version of the manuscript. C. Comas was supported during 2009–2010 by a “Juan de la Cierva” contract from the Spanish Government. This research has been supported by the Spanish Ministry of Education and Science (MTM2007-62923).

References

  1. Baddeley A, Møller J (1989) Nearest-neighbour Markov point processes and random sets. Int Stat Rev 57:89–121CrossRefGoogle Scholar
  2. Baddeley A, Turner R (2000) Practical maximum pseudolikelihood for spatial point patterns (with discussion). Aust N Z J Stat 42:283–322CrossRefGoogle Scholar
  3. Baddeley A, van Lieshout MNM (1993) Stochastic geometry in high-level vision. In: Mardia KV, Kanji GK (eds) Statistics and images, vol 1. Advances in applied statistics. Carfax Publishing, Abingdon, pp 231–256Google Scholar
  4. Berman M, Turner R (1992) Approximating point process likelihoods with GLIM. Appl Stat 41:31–38CrossRefGoogle Scholar
  5. Besag JE (1974) Spatial interaction and the statistical analysis of lattice systems (with discussion). J R Stat Soc B 36:192–236Google Scholar
  6. Besag JE (1977) Some methods of statistical analysis for spatial data. Bull Int Stat Inst 47:77–92Google Scholar
  7. Comas C (2009) Modelling forest regeneration strategies through the development of a spatio-temporal growth interaction model. Stoch Environ Res Risk Assess 23:1089–1102CrossRefGoogle Scholar
  8. Comas C, Mateu J, Delicado P (2010) On tree intensity estimation for forest inventories: some statistical issues (submitted)Google Scholar
  9. Cressie N (1993) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  10. Degenhardt A (1999) Description of tree distribution and their development through marked Gibbs processes. Biom J 41:457–470CrossRefGoogle Scholar
  11. Degenhardt A, Pofahl U (2000) Simulation of natural evolution of stem number and tree distribution pattern in a pure pine stand. Environmetrics 11:197–208CrossRefGoogle Scholar
  12. Diggle PJ (1986) Parametric and non-parametric estimation for pairwise interaction point processes. In: Proceedings of the 1st world congress of the Bernoulli societyGoogle Scholar
  13. Diggle PJ (2003) Statistical analysis of spatial point patterns. Hodder Arnold, LondonGoogle Scholar
  14. Diggle PJ, Fiksel T, Grabarnik P, Ogata Y, Stoyan D, Tanemura M (1994) On parameter estimation for pairwise interaction point processes. Int Stat Rev 62:99–117CrossRefGoogle Scholar
  15. Fiksel T (1984) Estimation of parameterized pair potentials of marked and non-marked Gibbsian point processes. Electron Inform Kybernet 20:270–278Google Scholar
  16. Fiksel T (1988) Estimation of interaction potentials of Gibbsian point processes. Statistics 19:77–86CrossRefGoogle Scholar
  17. Gates DJ, Westcott M (1986) Clustering estimates for spatial point distributions with unstable potentials. Ann Inst Stat Math 38:123–135CrossRefGoogle Scholar
  18. Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell 6:721–741CrossRefGoogle Scholar
  19. Geyer CJ (1999) Likelihood inference for spatial point processes. In: Barndorff-Nielsen OE, Kendall WS, van Lieshout MNM (eds) Stochastic geometry: likelihood and computation. Chapman and Hall/CRC, Boca Raton, pp 79–140Google Scholar
  20. Geyer CJ, Møller J (1994) Simulation and likelihood inference for spatial point processes. Scand J Stat 21:359–373Google Scholar
  21. Geyer CJ, Thompson EA (1992) Constrained Monte Carlo maximum likelihood for dependent data (with discussion). J R Stat Soc B 54:657–699Google Scholar
  22. Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Wiley, New YorkGoogle Scholar
  23. Mateu J, Montes F (2001a) Likelihood inference for Gibbs processes in the analysis of spatial point patterns. Int Stat Rev 69:81–104CrossRefGoogle Scholar
  24. Mateu J, Montes F (2001b) Pseudo-likelihood inference for Gibbs processes with exponential families through generalized linear models. Stat Inference Stoch Process 4:125–154CrossRefGoogle Scholar
  25. Molina R, Ripley BD (1989) Using spatial models as priors in astronomical image analysis. J Appl Stat 16:193–206CrossRefGoogle Scholar
  26. Møller J (1999) Markov chain Monte Carlo and spatial point processes. In: Barndorff-Nielsen OE, Kendall WS, van Lieshout MNM (eds) Stochastic geometry: likelihood and computation. Chapman and Hall/CRC, Boca Raton, pp 141–172Google Scholar
  27. Møller J, Waagepetersen RP (2004) Statistical inference and simulation for spatial point processes. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  28. Moyeed RA, Baddeley A (1991) Stochastic approximation of the MLE for a spatial point pattern. Scand J Stat 18:39–50Google Scholar
  29. Ogata Y, Tanemura M (1981) Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Ann Inst Stat Math 33:315–338CrossRefGoogle Scholar
  30. Ogata Y, Tanemura M (1984) Likelihood analysis of spatial point patterns. J R Stat Soc B 46:496–518Google Scholar
  31. Penttinen A (1984) Modelling interaction in spatial point patterns: parameter estimation by the maximum likelihood method. Jyvaskyla studies in computer science, economics and statistics, vol 7. Cambridge University Press, CambridgeGoogle Scholar
  32. Penttinen A, Stoyan D, Henttonen HM (1992) Marked point processes in forest statistics. For Sci 38:806–824Google Scholar
  33. Renshaw E (2002) Two-dimensional spectral analysis for marked point processes. Biom J 44:1–28CrossRefGoogle Scholar
  34. Renshaw E, Comas C, Mateu J (2009) Analysis of forest thinning strategies through the development of space-time growth-interaction simulation models. Stoch Environ Res Risk Assess 23:275–288CrossRefGoogle Scholar
  35. Ripley BD (1976) The second-order analysis of stationary point processes. J Appl Probab 13:255–266CrossRefGoogle Scholar
  36. Ripley BD (1981) Spatial statistics. Wiley, New YorkCrossRefGoogle Scholar
  37. Ripley BD (1988) Statistical inference for spatial processes. Cambridge University Press, CambridgeGoogle Scholar
  38. Särkkä A (1995) Pseudo-likelihood approach for Gibbs point processes in connection with field observations. Statistics 26:89–97CrossRefGoogle Scholar
  39. Särkkä A, Tomppo E (1998) Modelling interactions between trees by means of field observations. For Ecol Manag 108:57–62CrossRefGoogle Scholar
  40. Stoyan D, Penttinen A (2000) Recent applications of point process methods in forestry statistics. Stat Sci 15:61–78CrossRefGoogle Scholar
  41. Stoyan D, Stoyan H (1994) Fractals, random shapes and point fields: methods of geometrical statistics. Wiley, ChichesterGoogle Scholar
  42. Stoyan D, Stoyan H (1996). Estimating pair correlation function of planar cluster processes. Biom J 38:259–271CrossRefGoogle Scholar
  43. Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications. Wiley, New YorkGoogle Scholar
  44. Strauss DJ (1975) A model for clustering. Biometrika 62:467–475CrossRefGoogle Scholar
  45. Strauss DJ (1986) On a general class of models for interaction. SIAM Rev 28:513–527CrossRefGoogle Scholar
  46. Takacs R (1983) Estimator for the pair potential of a Gibbsian point process. Johannes Kepler Universitat Linz, AustriaGoogle Scholar
  47. Takacs R (1986) Estimator for the pair potential of a Gibbsian point process. Math Oper Stat Ser Stat 17:429–433Google Scholar
  48. Tomppo E (1986) Models and methods for analysing spatial patterns of trees. Communicationes Instituti Forestalis Fenniae 138:1–65Google Scholar
  49. Vanclay JK (1994) Modelling forest growth and yield: application to mixed tropical forest. CAB International, UKGoogle Scholar
  50. van Lieshout MNM, Baddeley A (1995) Markov chain Monte Carlo methods for clustering of image features. In Proceedings of the fifth international conference on image processing and its applications, vol 410. IEE Conference Publication, London, pp 241–245Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversitat Jaume ICastellónSpain

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